Admissions Mathematics PhD Admissions from BS in EE??

Basically, my heart is set on studying mathematics at the doctoral level and I am trying to figure out what I need to do to get there/what tier of schools I realistically could be a competitive applicant for.

I'm an Electrical Engineer from a state university that graduated with a 3.5 GPA in May 2016. However, my true interest has always been in mathematics. As an undergrad, I took Introductory Real Analysis 1 and Intro. Abstract Algebra 1 and did well in both courses.

So here's the thing, I actually planning on taking a semester of graduate level math courses in the Fall of this year (graduate Topology, Real Analysis 1, and Abstract Algebra 1). I'm leaving my day job and I've actually already registered at a local college as a non degree seeking student(somewhat reputable program). Some people think I'm crazy for ramping up the difficulty that quickly, but I've been studying on my own and actually feel like I can handle this course load in a semester. Plus I already know some graduate topology.

My letters of recommendation are going to be fairly strong, one coming from someone who received their PhD under Robrion Kirby.

With all this in mind, where do I fit into the picture if

1) I do well in the grad courses I'm planning on taking
2)My letter of recommendations end up being strong
3) I don't bomb the GRE ( a big if)

So my question is, with all this in mind.

Do I have a chance at a school like Duke? or Texas A&M or do I need to go WAAAY down the list?

P.S I ideally would like to study something like computational topology (why I'm interested in Duke).

You might want to look at the undergraduate math course offerings of any proposed school and see how much of it you have studied (either formal course or self-study). This might help to give you an idea of where they expect their own BS/BA graduates to be, so you can evaluate your position against them. If you do this, be very honest about your evaluations.

You might want to look at the undergraduate math course offerings of any proposed school and see how much of it you have studied (either formal course or self-study). This might help to give you an idea of where they expect their own BS/BA graduates to be, so you can evaluate your position against them. If you do this, be very honest about your evaluations.

While I'll be missing courses like Differential Geometry, Advanced Linear Algebra, Intro Complex Analysis etc. I will have much harder courses like Modern Algebra 1, and, particularly, measure theoretic Real Analysis. These are graduate level courses pretty much anywhere you go, including Duke.

Keep in mind that I will have a variety of applied courses in programming and Electrical Engineering that are definitely relevant for studying applied mathematics, which is what I want to do. .

I guess, I was hoping that someone on here might have some specific experience with applying to mathematics graduate programs or evaluating applications. Specific experience that would allow them to tell me "Are you crazy? There is no way you would get into Duke", or "Wow if you pull that off, you might have a shot".